The value of $$\sum_{k = 1}^{13} \frac{1}{\sin\left(\frac{\pi}{4} + \frac{(k - 1)\pi}{6}\right)\sin\left(\frac{\pi}{4} + \frac{k \pi}{6}\right)}$$ is equal to
$$3 - \sqrt{3}$$
$$2(3 - \sqrt{3})$$
$$2(\sqrt{3} - 1)$$
$$2(2 + \sqrt{3})$$
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